According to Proposition 5.5 of [1]}, we also need to prove \(\lim \limits _{n\rightarrow \infty }\dfrac{1}{n}\sum _{i=1}^{\infty }\mathbb {E}\lbrace \psi _i(\mathbf {Y}_i,\mathbf {\theta })\rbrace =0\)  implies \( \mathbf {\theta }=\mathbf {\theta }_0\) . Let \(\mathbf {\theta }_{10} = \mathbf {\beta }_0, \mathbf {\theta }_{20} = \mathbf {\Sigma }_{\mathbf {\varepsilon }0},\mathbf {\theta }_{30} = \mathbf {\Sigma }_{\mathbf {b}0}\)  denote the true parameter value, and \(\mathbf {\Sigma }_{i0} = I_{J_i}\otimes \mathbf {\theta }_{20} + \mathbf {A}_i\mathbf {\theta }_{30}\mathbf {A}_i^T\) . Taking expectation of \(\psi _i(\mathbf {Y}_i, \mathbf {\theta })\) , we have
\(\mathbb {E}\lbrace \psi _{i1}(\mathbf {Y}_i,\mathbf {\theta })\rbrace  = \mathrm {vec}^T\lbrace (\mathbf {\theta }_{10} - \mathbf {\beta })\mathbf {X}_i\rbrace {\mathbf {\Sigma }}_{i}^{-1}(\mathbf {X}_i\otimes I_r),\) 
\(\mathbb {E}\lbrace \psi _{i2}(\mathbf {Y}_i,\mathbf {\theta })\rbrace  = \dfrac{1}{2}\mathrm {vec}^T({\mathbf {\Sigma }}_{i0}^{-1}\mathbf {\Sigma }_i{\mathbf {\Sigma }}_{i0}^{-1} - \mathbf {\Sigma }_i^{-1})\mathbf {E}_{J_ir}\mathbf {M}_{i1},\) 
\(\mathbb {E}\lbrace \psi _{i3}(\mathbf {Y}_i,\mathbf {\theta })\rbrace  = \dfrac{1}{2}\mathrm {vec}^T({\mathbf {\Sigma }}_{i0}^{-1}\mathbf {\Sigma }_i{\mathbf {\Sigma }}_{i0}^{-1} - \mathbf {\Sigma }_i^{-1})\mathbf {E}_{J_ir}\mathbf {M}_{i2}.\) 
